Key Concepts and Formulas

1. Cofactor Expansion Theorem: For an $n \times n$ matrix $A$ with elements $a_{ij}$ and cofactors $A_{ij}$:

   • Property 1 (Expansion along a row/column): The sum of the products of the elements of any row (or column) with their corresponding cofactors is equal to the determinant of the matrix $A$. Mathematically, for expansion along row $i$: $\sum_{k=1}^{n} a_{ik} A_{ik} = |A|$.
   • Property 2 (Expansion using cofactors of a different row/column): The sum of the products of the elements of any row (or column) with the cofactors of a different row (or column) is always zero. Mathematically, for elements of row $i$ multiplied by cofactors of row $j$ (where $i \neq j$): $\sum_{k=1}^{n} a_{ik} A_{jk} = 0$.

2. Logarithm Properties: These properties are essential for simplifying the elements of matrix $A$:

   • $\log_b a^n = n \log_b a$
   • $\log_{b^m} a^n = \frac{n}{m} \log_b a$
   • $\log_b a \cdot \log_a b = 1$ (This is a direct consequence of the change of base formula).

3. Determinant of a $2 \times 2$ Matrix: For a matrix $M = \begin{bmatrix} p & q \\ r & s \end{bmatrix}$, its determinant is $|M| = ps - qr$.

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Step-by-Step Solution

Step 1: Understand the definition of $C_{ij}$ and its relation to the Cofactor Expansion Theorem. The problem defines the elements of matrix $C$ as $C_{ij} = \sum_{k=1}^{2} a_{ik} A_{jk}$. This expression directly corresponds to the Cofactor Expansion Theorem. Let's analyze the elements of $C$ based on the theorem:

• For $C_{11}$ (where $i=1, j=1$): $C_{11} = \sum_{k=1}^{2} a_{1k} A_{1k} = a_{11}A_{11} + a_{12}A_{12}$. According to Property 1 of the Cofactor Expansion Theorem, this sum is equal to the determinant of $A$, i.e., $C_{11} = |A|$.

• For $C_{12}$ (where $i=1, j=2$): $C_{12} = \sum_{k=1}^{2} a_{1k} A_{2k} = a_{11}A_{21} + a_{12}A_{22}$. Here, the elements are from the first row ($a_{1k}$), but the cofactors are from the second row ($A_{2k}$). Since $i \neq j$ (i.e., $1 \neq 2$), according to Property 2 of the Cofactor Expansion Theorem, this sum is zero. Thus, $C_{12} = 0$.

• For $C_{21}$ (where $i=2, j=1$): $C_{21} = \sum_{k=1}^{2} a_{2k} A_{1k} = a_{21}A_{11} + a_{22}A_{12}$. Similarly, the elements are from the second row ($a_{2k}$), but the cofactors are from the first row ($A_{1k}$). Since $i \neq j$ (i.e., $2 \neq 1$), this sum is zero by Property 2. Thus, $C_{21} = 0$.

• For $C_{22}$ (where $i=2, j=2$): $C_{22} = \sum_{k=1}^{2} a_{2k} A_{2k} = a_{21}A_{21} + a_{22}A_{22}$. This sum involves elements of the second row ($a_{2k}$) and their corresponding cofactors ($A_{2k}$). By Property 1, this sum is equal to the determinant of $A$, i.e., $C_{22} = |A|$.

From this analysis, we can deduce the structure of matrix $C$: $C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} = \begin{bmatrix} |A| & 0 \\ 0 & |A| \end{bmatrix}$ This crucial insight simplifies the problem significantly, as we only need to calculate $|A|$.

Step 2: Simplify the elements of Matrix A using Logarithm Properties. The given matrix is $A = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix}$. Let's simplify each element:

• $a_{11} = \log_5 128 = \log_5 (2^7) = 7 \log_5 2$
• $a_{12} = \log_4 5 = \log_{2^2} 5 = \frac{1}{2} \log_2 5$
• $a_{21} = \log_5 8 = \log_5 (2^3) = 3 \log_5 2$
• $a_{22} = \log_4 25 = \log_{2^2} (5^2) = \frac{2}{2} \log_2 5 = \log_2 5$

Now, the simplified matrix $A$ is: $A = \begin{bmatrix} 7 \log_5 2 & \frac{1}{2} \log_2 5 \\ 3 \log_5 2 & \log_2 5 \end{bmatrix}$

Step 3: Calculate the Determinant of A, $|A|$. Using the formula for a $2 \times 2$ matrix determinant ($|A| = a_{11}a_{22} - a_{12}a_{21}$): $|A| = (7 \log_5 2)(\log_2 5) - \left(\frac{1}{2} \log_2 5\right)(3 \log_5 2)$ We apply the logarithm property $\log_b a \cdot \log_a b = 1$: $|A| = 7 (\log_5 2 \cdot \log_2 5) - \frac{3}{2} (\log_2 5 \cdot \log_5 2)$ $|A| = 7(1) - \frac{3}{2}(1)$ $|A| = 7 - \frac{3}{2}$ To combine these terms, we find a common denominator: $|A| = \frac{14}{2} - \frac{3}{2} = \frac{11}{2}$ So, the determinant of matrix $A$ is $\frac{11}{2}$.

Step 4: Determine the elements of Matrix C. From Step 1, we established that $C = \begin{bmatrix} |A| & 0 \\ 0 & |A| \end{bmatrix}$. Substituting the value of $|A|$ we just calculated: $C = \begin{bmatrix} \frac{11}{2} & 0 \\ 0 & \frac{11}{2} \end{bmatrix}$

Step 5: Calculate the Determinant of C, $|C|$. For matrix $C = \begin{bmatrix} \frac{11}{2} & 0 \\ 0 & \frac{11}{2} \end{bmatrix}$, its determinant is: $|C| = \left(\frac{11}{2}\right)\left(\frac{11}{2}\right) - (0)(0)$ $|C| = \frac{121}{4} - 0$ $|C| = \frac{121}{4}$

Step 6: Calculate the final expression $8|C|$. The problem asks for the value of $8|C|$: $8|C| = 8 \times \frac{121}{4}$ $8|C| = 2 \times 121$ $8|C| = 242$

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Common Mistakes & Tips

• Ignoring the Cofactor Expansion Theorem's second property: A common mistake is to explicitly calculate all cofactors $A_{ij}$ for matrix $A$ and then perform the summations for each $C_{ij}$. This is very time-consuming for a $2 \times 2$ matrix and misses the elegant shortcut provided by the theorem, which immediately tells us $C_{12}=0$ and $C_{21}=0$.
• Errors in Logarithm Simplification: Ensure correct application of logarithm rules, especially changing bases or handling powers within the logarithm. Simplifying elements of $A$ at the beginning makes subsequent calculations much cleaner.
• Arithmetic Errors: Double-check calculations, especially when dealing with fractions.

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Summary

This problem effectively tested the understanding of the Cofactor Expansion Theorem in determinants, combined with fundamental logarithm properties. The most critical step was recognizing that the definition of $C_{ij}$ directly maps to the properties of cofactor expansion. This insight revealed that matrix $C$ is a diagonal matrix where its diagonal elements are equal to the determinant of $A$, i.e., $C = \text{diag}(|A|, |A|)$. After simplifying the logarithmic terms in matrix $A$ and calculating $|A|$, we found $|A| = \frac{11}{2}$. Subsequently, $|C| = \left(\frac{11}{2}\right)^2 = \frac{121}{4}$, leading to the final answer $8|C| = 242$.

The final answer is $\boxed{242}$, which corresponds to option (D).